Characters of irreducible representations

In Section 6.1 we define irreducible representations. Then we state, prove and illustrate Schur s lemma. Schur s lemma is the statement of the all-or-nothing personality of irreducible representations. In the Section 6.2 we discuss the physical importance of irreducible representations. In Section 6.3 we introduce invariant integration and apply it to show that characters of irreducible representations form an orthonormal set. In the optional Section 6.4 we use the technology we have developed to show that finite-dimensional unitary representations are no more than the sum of their irreducible parts. The remainder of the chapter is devoted to classifying the irreducible representations of 5(7(2) and 50(3). 

According to the properties of characters of irreducible representations as components of vectors (4.3-8), we obtain... 

Note first that y(E) = 6 while all other characters are zero. The reason is that the operation E transforms each into itself while every rotation operation necessarily shifts every 0, to a different place. Clearly this kind of result will be obtained for any /z-membered ring in a pure rotation group C . Second, note that the only way to add up characters of irreducible representations so as to obtain y = 6 for E and y - 0 for every operation other than E is to sum each column of the character table. From the basic properties of the irreducible representations of the uniaxial pure rotation groups (see Section 4.5), this is a general property for all C groups. Thus, the results just obtained for the benzene molecule merely illustrate the following general rule ... 

TABLE 4.7 Properties of Characters of Irreducible Representations in Point Groups... 

Any textbook on application of group theory in molecular spectroscopy contains tables of characters of irreducible representations, which correspond to various symmetry groups of molecules. 

Now, let us find (p. 920) how many times [a(a)] the irreducible representation a is present in the reducible representation (the sum over classes number of operations in class x the calculated character x the character of irreducible representation) ... 

Equation (4.142) is a consequence of the orthogonality of the characters of irreducible representations of the translation group to the character of the unit representation (fe = 0), while (4.143) means that the characters of a regular representation of the group are equal to zero for aU elements of the group except for the identity element i.e. except for the identity translation and equivalent translations through the superlattice vectors A). 

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